Since the functional \(J_\rho \)  on the Hilbert space \(H\)  is strongly indefinite, and the nonlinearity is exponential in the negative direction, it is desirable to restrict it to a nice Nehari-type submanifold of \(H\) , so that we can get better control in the negative part of the functional, as in [1]}, [2]}.
This can be done in the spirit of [3]}, [4]}.
Namely, consider the map
\(G=(G_1, G_2)\colon \left(H^1(M)\right)^2 \times \left(H^{\frac{1}{2}}(\Sigma M)\right) \rightarrow \left(H^{\frac{1}{2},-}(\Sigma M)\right)^2\) 
